Advice on Reading solutions

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guptaamitu1

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guptaamitu1

#1 h Aug 4, 2022, 7:08 PM • 4 Y

Y by truffle, HamstPan38825, samrocksnature, megarnie

I was recently facing some problems regarding reading solutions. I have read some blogs of Evan Chen on that (1. For solution reading ; 2. General) and have got advice from some other expert people too. What I mainly understood is the following:

It is not necessary to understand every possible solution to a problem. One should look at a solution only if it is

fairly clean/short
and idea behind it is very clever/unique

In general looking at solutions posted by some famous people suffices. After reading a solution, one should mainly try to understand the key idea behind it, from which the rest of the solution could easily be reconstructed (called "crossing the ocean" thing by Evan). Sometimes a solution might seem long/messy but main idea/structure of it might be pretty neat. Also, it would be quite counter-productive if seeing the sol takes too much time, most of the gain still comes from attempting the questions on your own.

But I have two doubts now:

1. Suppose I see a long solution. Then maybe I should first try to understand it properly, like understand all its arguments precisely. Then I should try to find the key idea behind the solution. But this process might take some time. Sometimes it might even take 20-30 minutes or so. So is this time worth spending? Since in general it is said that "It's going to be very counter-productive if seeing the sol takes too much time, most of the gain still comes from attempting the questions on your own."

2. Not all problems have official solution or some famous people posted solution to them. In that case mainly which solutions should I focus more on? Like the ones which seem shorter in length or ones which are on post #2? Of course the latter classification doesn't seem to be a nice idea. On the first classification, sometimes a long solution might have a short key idea and similarly, a short-looking solution might be harder to grasp.

Two recent examples of doubt 1. are as follows:

The following problem is Sharygin Finals 2018 Grade 10 P4

Sharygin Finals 2018 Grade 10 P4 wrote:

We say that a finite set $S$ of red and green points in the plane is orderly if there exists a triangle $\delta$ such that all points of one colour lie strictly inside $\delta$ and all points of the other colour lie strictly outside of $\delta$ . Let $A$ be a finite set of red and green points in the plane, in general position. Is it always true that if every $1000$ points in $A$ form a orderly set then $A$ is also orderly?

Following is its official solution.

The solution seems long and complicated. But after I read it, I understood that the overall idea behind it is quite clever. One doubt is the following:

Should I be trying to verify some of the facts written in the solution without proof, for example why the mentioned construction worked and why everything written in the following paragraph is true:

The second example is Sharygin Finals 2018 Grade 9 P8.

Sharygin Finals 2018 Grade 9 P8 wrote:

Consider a fixed regular $n$ -gon of unit side. When a second regular $n$ -gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.

$[asy] int n=9; draw(polygon(n)); for (int i = 0; i<n;++i) { draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4)); draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2)); } [/asy]$

Let $A$ be the area of a regular $n$ -gon of unit side, and let $B$ be the area of a regular $n$ -gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$ .

Below is the official solution:

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Math4Life7

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Math4Life7

#2 h Aug 4, 2022, 7:12 PM • 1 Y

Y by NegativeZeroPlusOne

as a person who is terrible at reading solutions and still decently okay at comp math. I think solution reading isn't too important you just need to understand the concept in one way or another not necesarrily through solutions tho

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guptaamitu1

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guptaamitu1

#3 h Aug 4, 2022, 7:44 PM • 3 Y

Y by Mango247, Mango247, Mango247

Math4Life7 wrote:

as a person who is terrible at reading solutions and still decently okay at comp math.

My question was not regarding comp math. I meant to ask about olympiad math.

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BakedPotato66

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BakedPotato66

#4 h Aug 4, 2022, 10:16 PM

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I guess one thing that could be helpful is watching video solutions (if those exist for olympiad math, for example, probably Evan Chen has some on his YT channel). bc I think video solutions are much easier to understand and follow along

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anurag27826

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anurag27826

#5 h Aug 5, 2022, 11:57 AM

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BakedPotato66 wrote:

I guess one thing that could be helpful is watching video solutions (if those exist for olympiad math, for example, probably Evan Chen has some on his YT channel). bc I think video solutions are much easier to understand and follow along

Every oly problem doesn't have a video solution.

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ZETA_in_olympiad

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ZETA_in_olympiad

#6 h Aug 5, 2022, 12:14 PM

Y by

guptaamitu1 wrote:

Math4Life7 wrote:

as a person who is terrible at reading solutions and still decently okay at comp math.

My question was not regarding comp math. I meant to ask about olympiad math.

Yeah, comp math can be computational. But according to me, still what M4L7 says it true.

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HamstPan38825

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HamstPan38825

#8 h Aug 5, 2022, 5:06 PM • 12 Y

Y by Bradygho, metricpaper, Mogmog8, eagles2018, peace09, guptaamitu1, akasht, ETS1331, eibc, rama1728, third_one_is_jerk, bestzack66

There's something which I like to call the Uncertainty Principle for olympiads. The interpretation, well, is taken quite literally.

Of course, much of this is based on (comparatively, scant) personal experience. But I think I've thought about this problem enough that I have a few things to say.

In fact, the continuum of your comfortability with any certain problem is more discrete than continuous. I will carry over Evan's notation, as it makes many things easier to explain.

Suppose we have a problem $P$ .

First, I personally do not like it when $P$ is arbitrary. Note that everything that follows is cast under the uncertainty principle: the more you try to concretely classify a problem, the less you perceive the structure of it past a certain extent. There are essentially three classes of problems:

Exercises: spinoffs of a problem, which illustrate an obvious idea or principle. Textbook examples, theoretical illustrations, etc. also fall under this category.
Practices: the idea is essentially given (e.g. "practice on the use of projective transformations"), but the problems are likely full-fledged olympiad problems with their own developed structure.
Problems: an arbitrary problem about which you predetermine nothing before you see it. In other words, akin to actually solving olympiad problems.

The second category is by far the most important in training, by an overwhelming supermajority. I think Evan's 2013/2014 platitude posts are somewhat outdated: nowadays, the amount of resources, categorizations, structured books and handouts, etc. is growing exponentially. Capitalize off the work that has been done already for you. Don't do arbitrary problems all the time.

Of course, this is not actually directly related to the topic of solutions themselves, but it will be pivotal to my main point. First, let us consider the first category of problems.

This is the type of problem that either seems immediately trivial or it doesn't. As an example, say $P$ is a preliminary exercise in a book designed to introduce Hall's Marriage Lemma. A common misperception is as follows:

Misperception wrote:

I cannot really spend my time well on this problem because I know it is trivial by something and there will be nothing else to find.

This is totally false: the example of Hall is quite extreme, but this applies in many other places as well. You will learn something by thinking about a problem that would lead to a result $R$ without even coming close to finding $R$ .

For example, if you have an idea, go down that path and just keep going until you get hopelessly stuck; then, if you have nowhere else to go, read the solution and learn about the proposed idea. You may not be able to make the connection why did I get stuck? immediately.

Some people like to note down literally every detail of every problem: okay, I got stuck on this problem, come back and review it in $n$ days and try to think about why I got stuck, or contriving some kind of reason to explain it. This is simply stupid: artificially synthesized intuition is an oxymoron. If you can immediately see why you got stuck, great! If not, well, keep your scratch work/notes somewhere, understand the correct solution (write it down), and move on.

Though I've explained this in the specific context of exercises, some of the above also applies to general problems and practices as well. Again, the uncertainty principle: there is never an explicit formula, and intuitive exceptions always override established systems of thought.

Now onto the second category; I think this is where I can begin to address your doubts explicitly. Much of the philosophy behind reading solutions is derived from the following principle, which is similar to the one I explained before:

principle wrote:

If any bit of non-concrete intuition is not obvious to you, it does not exist.

Suppose you're working on an angle chasing section of geometry and one of the solutions you find explains why we should construct the point $P$ because there is an incenter Miquel configuration hidden in the problem, yada yada, and they're acclaimed for making that observation, say on the AoPS thread.

If you've learned about this specific configuration before, and given the statement, you immediately realize you missed that, well, great! The point, though, is that what you're training yourself to do is angle chase in an arbitrary diagram correctly in an analytical fashion. If you don't know the configuration, well, it doesn't mean the problem is above you. In some sense, reading these solutions is actually detrimental, as I will demonstrate.

Let's use the same angle chasing problem with the construction of the point $P$ , and say that there is a solution that is simply vanilla angle chasing (that also uses the construction). Assume further that you missed the construction, which was the key point and made all the other correct observations. Then, by reading the angle chasing solution, you find that you should have constructed $P$ because [insert specific *concrete* implications for angle chasing here; i.e. creates a cyclic quadrilateral, encompasses a concurrency, etc.]. You may not know exactly what the motivation was, but that's fine! If you immediately understood why constructing $P$ would solve the problem, your brain has automatically created an intuitive "picture" of the problem. If there exists another problem for which having the intuition of this problem will make it much easier, your brain *will* recognize something familiar. Pondering for 30 minutes over why is a complete waste of those 30 minutes.

Clean and short solutions are not exactly the best solutions. In fact, I have heard the following quote more than once:

Quote:

When written correctly, any inequality has a reasonable legible one-line solution.

There will always be weird solutions. Honorable mention that I have to include: spoilers for IMO 2001/2
I suppose it's a clever idea. No freaking way I'm coming up with that in contest though.

There is no objectively "better" solution. Your work may resemble that of a laughably overkill solution, or follow the lines of a remarkably inefficient one. At the end of the day, it doesn't matter. In some ways, there is a net gain with respect to time: it is easier to complete a 1000-piece puzzle with 250 pieces missing than it is to complete a 250-piece puzzle. Maybe one day you will come back to the problem and find the 250-piece solution.

Problems are, ultimately, their solutions, not themselves. The same problem (for example, 2011 USAJMO/5) can be spoilers Spend a little time scrolling through to find solutions that resemble your work to the greatest extent; this should be your first priority if and only if there is not an official solution in the context of where the practice problem appeared (for example, selected solutions in a textbook.) If that solution exists, you should read it no matter what: it either resembles your solution completely, or it illustrates an application of the idea you were trying to learn that you missed. This is why problems in this category are so useful: you're guaranteed to take away at least something concrete from the problem (from reading the official solution), and doing problems with similar flavor consciously helps your brain build that "abstract intuition" stronger and stronger.

Hopefully the above discourse has illustrated why I am against doing arbitrary problems: even if those problems are from well-known or affluent contests (IMO shortlist, USA contests, etc.) You will find yourself missing 250 pieces of a 1000 piece puzzle that can be completed much, much less than you will find yourself missing 750 pieces into a puzzle that cannot be completed. It may be frustrating to make *so much progress* on one approach and have none of the complete solutions take your path, instead diverging in many different directions none of which you even considered. But it happens: you want to have something to fall back on.

Thus, embrace official solutions in context; they're your best friend.

Of course, this is not to say that you should ignore all solutions entirely. If you cannot find a solution that traces your own work, you can simply just read a solution to the problem that feels easy to understand (intuitively). If many of the solutions go in one specific direction that differs from your original approach, do some work in that direction. You don't stop thinking about the problem the moment you start reading any solution. Keep your brain active: try to follow a solution that is structured in a way you can perceive. If the first solution you find doesn't satisfy this, look for another one. If all things fail (or you decide consciously to do so), simply drop the problem and come back to it later.

Essentially, the above paragraph can be summarized in one sentence: make it work.

Above all, we truly realize how to think about attempting problems, looking at solutions, and using hints when we realize that there is nothing to think about.

This post has been edited 1 time. Last edited by HamstPan38825, Aug 5, 2022, 5:14 PM

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Math4Life7

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Math4Life7

#9 h Aug 5, 2022, 6:49 PM • 1 Y

Y by NegativeZeroPlusOne

guptaamitu1 wrote:

Math4Life7 wrote:

as a person who is terrible at reading solutions and still decently okay at comp math.

My question was not regarding comp math. I meant to ask about olympiad math.

I was talking about competition math but I dont have too much experience with oly so...

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guptaamitu1

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guptaamitu1

#10 h Aug 6, 2022, 10:54 AM

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@HamstPan38825 , Thanks. That was quite helpful.

I think you gave emphasis on the fact that it much better to try practice problems. But sometimes, I lag resources for that. Like for Geometry, one can find lot of nice material online. But when it comes to other topics (like Algebra, Combinatorics, and Number Theory), then I am unable to find nice resources. There might be some introductory type resources, but say if one is trying to get better at tackling IMO 3/6, I simply don't know exactly which handouts/books to refer.

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anurag27826

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anurag27826

#11 h Aug 6, 2022, 12:53 PM • 1 Y

Y by Mango247

guptaamitu1 wrote:

But when it comes to other topics (like Algebra, Combinatorics, and Number Theory), then I am unable to find nice resources. There might be some introductory type resources, but say if one is trying to get better at tackling IMO 3/6, I simply don't know exactly which handouts/books to refer.

True, There is a great need for a Combinatorics and NT Marathon just like the active Geo Marathon.

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HamstPan38825

8857 posts

HamstPan38825

#12 h Aug 7, 2022, 2:53 PM • 1 Y

Y by guptaamitu1

guptaamitu1 wrote:

@HamstPan38825 , Thanks. That was quite helpful.

I think you gave emphasis on the fact that it much better to try practice problems. But sometimes, I lag resources for that. Like for Geometry, one can find lot of nice material online. But when it comes to other topics (like Algebra, Combinatorics, and Number Theory), then I am unable to find nice resources. There might be some introductory type resources, but say if one is trying to get better at tackling IMO 3/6, I simply don't know exactly which handouts/books to refer.

I don't want to say anything too decisive here, as I'm nowhere near the level of IMO 3/6, and towards the most difficult end of olympiads there is really a luck-based aspect: a highly experienced contestant with a reasonable amount of training faced with a reasonable olympiad may find it possible to guarantee, say, 35+, but a guaranteed 42 is near impossible. (A similar idea plays out throughout all nontrivial contest levels.)

That being said, you're not asking about how to solve IMO 3/6's, which makes the above tangent somewhat irrelevant. I'm not sure how difficult or how unconventional you're trying to go for here, but some kinds of resources definitely exist: for example, to cite probably the most famous of which, XYZ Press's Topics in Functional Equations, Lemmas in Olympiad Geometry, and Number Theory: Concepts and Problems form a quite formidable trio.

Another piece of advice that is useful at all levels is to capitalize off the experiences of others, quite directly. I'm not a good person to give you specific guidance here: but if you ask a specific, concrete question (example: "can anyone give a good set of problems for practicing $\nu_p$ in binomial coefficients?" rather than "does anyone have any good recommendations for IMO 3/6 level $\nu_p$ problems?), there is likely someone who could help you. In general, you will be much better off with your solution-reading dilemma if there exists a person you know who has attempted the problem recently, still has it fresh on their mind, and can explain it to you and point out anything you've missed.

This post has been edited 1 time. Last edited by HamstPan38825, Aug 7, 2022, 2:53 PM

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